If you look at a square wave with amplitude of +/- A on a graph, you will see that it covers a range of 2A (A- -A=2A). This wave has a amplitude that fluctuates between these two values. However, an alternate wave with amplitude 2A has the same range (2A), yet it fluctuates between 0 and 2A. The two are identical in their form, they only differ how values are assigned.

Your statement explicitly says x(t) is an amplitude.
So t is the location on the x-axis and x(t) is a sample point value on the y-axis.
Adding a constant to x(t) just shifts the waveform up or down on the y-axis.

If you want to shift the wave on the x-axis then you would write something like x(t-n).
This has no overall effect on the y-axis amplitude excursions.

At any rate, if this is just a math exercise in shifting coordinates, a signal can be shifted horizontally by one's choice of zero time. Looking back at message #4, we can say either the signal goes from -5 sec to +5 sec, or 0 to 10 sec.

At any rate, if this is just a math exercise in shifting coordinates, a signal can be shifted horizontally by one's choice of zero time. Looking back at message #4, we can say either the signal goes from -5 sec to +5 sec, or 0 to 10 sec.

This assumes I'm interpreting alextsipkis's statements correctly.

I agree with you about the coordinate shifting part, you can do both.
Seems to me that the OP defined x to be a square wave generator of amplitude (+A, -A).
That gives A the definition of amplitude.
Unless A got redefined somewhere, I don't see how you could get a time shift by adding a voltage value to the square wave.
Did I miss a redefinition somewhere?
If so, I'm not seeing it, but I've been known to be blind sometimes.

Seems to me that the OP defined x to be a square wave generator of amplitude (+A, -A).
That gives A the definition of amplitude.
Unless A got redefined somewhere, I don't see how you could get a time shift by adding a voltage value to the square wave.
Did I miss a redefinition somewhere?
If so, I'm not seeing it, but I've been known to be blind sometimes.

No, adding a voltage can't produce a time shift. It's simply a matter of defining where "zero" is. Whether it's voltage or time, 0V or 0 sec is an arbitrary reference.

If I have a graph that covers a 10 sec range, from 0 to 10 sec, you might ask "when does t=0". And I might answer that t is zero at 2:00 p.m. today. But we could just as easily say t is zero at 5 seconds past 2:00 p.m., in which case the graph would run from -5 to +5 sec.

Offsetting the signal on the amplitude axis introduces a 0 Hz component in the frequency domain.

Shifting the signal on the time axis will introduce a phase change of frequency amplidudes. If you are transforming into sines and cosines the relative amplitudes for at any given frequency will vary inversily. (actually as a^2+b^2=constant, but that may be too much detail)